**Elementary Algebra – Chapter
3**

** **

**Definition: **A **rectangular
coordinate system** (Cartesian coordinate system) consists of two
perpendicular number lines. One number
line is drawn horizontally and the other is drawn vertically.

**Definition:** The horizontal number line is usually called
the **x-axis**.

**Definition:** The vertical number line is usually called
the **y-axis**.

**Definition:** The point of intersection of the two number
lines is called the **origin** of the coordinate
system.

**Definition:** The two axes form a coordinate plane and divide
it into **four quadrants** named Quadrant
I, Quadrant II, Quadrant III, and Quadrant IV as shown in the diagram here.

**Definition:** Each point in a coordinate plane can be identified
by a pair of real numbers x and y written in the form (x, y). The first number is called the **x-coordinate** and represents a number on the horizontal
number line. The second number is
called the **y-coordinate **and represents
a number on the vertical number line. The
numbers in the pair are called the **coordinate **of
the point.

**Definition:**** **A

**Definition: **The** General Form for the equation of a line is** Ax + By = C where A,
B, and C are real numbers and not both A and B are zero.

**Definition:** The **y-intercept**
of a line is the point (0, b) where the line intersects the y-axis.

**Definition:** The **x-intercept**
of a line is the point (a, 0) where the line intersects the x-axis.

**Remark:** The equation
y = b represents a horizontal line with y-intercept b.

**Remark:** The equation
x = a represents a vertical line with x-intercept a.

**Definition:** A **ratio**
is the quotient of two numbers.

**Definition:** Ratios that are used to compare quantities
with different units are called **rates**.

**Definition:** The slope of the non-vertical line through
two points (x_{1}, y_{1}) and (x_{2}, y_{2})
is

_{
}

**Remark:** Slope is
undefined for vertical lines.

**Remark:** Slope is
0 for horizontal lines

**From Section 3.5 The Slope-Intercept Form of the Equation of a Line**

If a linear
equation is written in the form **y = mx + b** where m and b are real numbers, the graph
of the equation is a line with slope m and y-intercept (0,b)

**Definition:** Two lines are **parallel**
if they have the same slopes and different y-intercepts.

**Definition:** Two lines are perpendicular if their slopes
are negative reciprocals of each other and conversely if the slopes of two
lines are negative reciprocals, the lines are perpendicular.

**Remark:** The fact
that two slopes m_{1} and m_{2} are negative reciprocals of
each other may be expressed algebraically in the each of the following ways.

**From Section 3.6 The Point-Slope Form of the Equation of a Line**

If a line with
slope m passes through the point (x_{1}, y_{1}) the equation
of the line is **y – y _{1} = m(x – x_{1})**.

**From Section 3.7 Graphing Linear Inequalities**

**Definition:**A linear inequality in two variables x and y is an inequality which can be written as

Ax + By < C, Ax + By > C, Ax + By = C, or Ax + By = C.

**Definition:** A point (x, y) is a solution of an inequality in two variables if the coordinates satisfy the inequality. (that is, if a true statement results when the coordinates are substituted for the variables in the inequality.)

**Definition:** The collection of all solutions of an inequality is called the solution set of the inequality.

**Definition:** The graph of an inequality is the set of points which are solutions of the inequality. (That is, the graph is the set of all points whose coordinates satisfy the inequality).

**Definition:** If the inequality symbol in an inequality in two variables is replaced with an equality symbol, the graph of the resulting equation is called the boundary line for the inequality.

**Fact:** The graph of an inequality in two variables is a half-plane.

**Fact:** The boundary line forms the boundary between the half-plane consisting of all solutions of the inequality and the half-plane consisting of all points which are not solutions of the inequality

**Procedure:** To graph a linear inequality in two variables:

a) Sketch the graph of the boundary line

i) as a solid line if the inequality symbol is either or .

ii) as a dashed line if the inequality symbol is either > or <.

b) Pick a point, not on the boundary line, as a test point and substitute its coordinates into the inequality.

c) If the result from Step b is a TRUE statement, the half-plane containing the test point is the solution set.

d) If the result from Step b is a FALSE statement, the half-plane which does not contain the test point is the solution set.

e) Shade the half-plane which is the solution and label all important points.